He uses the network to ridicule his perceived opponents, expand his political reach, shape news coverage, and otherwise deliver his unique style of rhetoric without filter from media adversaries or even his own advisors.

But with such unfettered access also comes danger. When presidents speak, markets often react. A chief executive, then, with a penchant for 3 a.m. missives may carry a whole new level of market risk.

"The world is a less, not more, certain place with a novice at the helm, and one who doesn't seem to mind shooting from the hip," David Rosenberg, chief economist and strategist at wealth management firm Gluskin Sheff, said in his daily market note Tuesday. "Remember, loose lips sink ships, and his willingness to continue to vent frustrations through his Twitter account is, in a word, bizarre (and tells me as a strategist that we are in a world of heightened political risk...)."

Others agree with Rosenberg's sentiment, contending that negative impressions about the president's reliability could make markets nervous.

"It would be difficult for anyone to look presidential when being reactionary on Twitter — full stop," said Art Hogan, chief market strategist at Wunderlich Securities, adding that he expected Trump aides could soon wrest Twitter account control from their boss because it would otherwise be "too easy to make mistakes and be reactionary."

'Walls and tariffs and trade wars'

But if Trump maintains control of his social media presence, investors may be in for a wild ride.

"The market reacts to the positive attributes of the president-elect — pro-growth, pro-business," Hogan said. "So the market would get nervous about the other side of the equation, where it's the more inflammatory things about — fill in the blank, there's plenty — walls and tariffs and trade wars."

Some experts, however, are unimpressed by the notion of a "tweet risk."

Marc Chandler, global head of currency strategy at Brown Brothers Harriman, pointed to stocks near record highs and widely held expectations for strong economic growth, brushing away the idea that Trump's predilection for Twitter has a discernible negative influence.

"It is a casino atmosphere, in keeping with Trump's area of expertise. I have found that it almost always pays after an election to ultimately do the opposite of what the markets initially priced in."-David Rosenberg, chief economist and strategist, Gluskin Sheff

Generally speaking, markets have demonstrated they're accustomed to the way politicians release opinions in the age of Twitter, Quincy Krosby, market strategist at Prudential Financial, told CNBC.

A tweet last year from Democratic presidential candidate Hillary Clinton influenced a major move in biotech during the campaign, and Trump's apparently impromptu Twitter vitriol has long been a known behavior — yet markets have gone higher.

"These are tools that are being incorporated by candidates. Whether they're young or old, they're getting their message out," Krosby said. "The market's job is to look through these things and get used to it, and the market gets bored with the same thing, so unless the tweets are really specific, and the tweets have major policy implications embedded in them, the markets will move on."

In other words, Krosby said that markets will "get used to his various ventings, his various moods," and will pay attention only to Trump's penchant for Twitter if it appears that he is speaking to concrete policy initiatives.

As for the market's post-election rise, count Rosenberg among those highly skeptical of the Trump rally. He worries that the billionaire businessman's ambitious promises won't do as much to push growth as the market is anticipating. Rosenberg compares Trump to Italy's Silvio Berlusconi, who rose to power also as a rich entrepreneur with big plans that ultimately came to little for his country.

The market has gotten ahead of itself in assuming Trump's plans to lower taxes and regulation and increase public spending will spark inflation, Rosenberg added.

"It is a casino atmosphere, in keeping with Trump's area of expertise," he said. "I have found that it almost always pays after an election to ultimately do the opposite of what the markets initially priced in."